Eternal black hole Hawking process

I am trying to understand the process of Hawking radiation in the case of an eternal (static/everlasting) black hole.

As a bit of background: i understand (semi-quantitatively) how one gets particles produced when one is a frame with constant acceleration. And I sort of understand how it arises in the case of a collapsing star (time dependent field can in principle always produce pairs, I think) but for the static case the boundary conditions confuse me a little.

In light of that I found a few papers by Unruh and so on which indicate that eternal black holes DO indeed produce Hawking pairs.

Thanks for the link, but I find his comment about the eternal BH case confusing.

Basically the question I want to ask is: what are the two frames/sets of modes one uses in the purely eternal case (i.e. static metric) and are Hawking pairs produced in this case? Do you just setup some modes on the past horizon and future infinity and transform between then?

discussing the Boulware vacuum. The author says that there is no pair creation around a static Schwarzschild black hole because there exists a static scalar vacuum (does this work with the No Hair theorem?)

Quotes/paraphrases from "Black Hole Physics" by Frolov and Novikov concerning the Boulware vacuum:

For a nonrotating black hole, the corresponding Green's function GB is a negative-frequency function on ℐ- and a positive-frequency function on ℐ+...

Consider a nonrotating spherical body of mass M and radius R0 slightly larger than 2M. Since the Killing vector ∂t is everywhere timelike, every particle has positive energy and particle creation is impossible...

The Green's function GB for a nonrotating black hole can be treated as the limit R0 → 2M. It has simple, regular behavior far from the black hole and corresponds to the absence of quantum radiation on both ℐ- and ℐ+, but reveals poor analytical behavior close to the event horizon. The renormalized quantities <B|Tμν|B> and <B|φ2|B> both diverge on H+ and H-.

Because of the presence of superradiant modes, GB is not well-defined for a rotating black hole.

Further comments from "Quantum Fields in Curved Space" by Birrell and Davies (Slightly different notation):

|0S> = Boulware vacuum. Modes are positive frequency wrt the Killing vector ∂t of Schwarschild time. They oscillate infinitely rapidly on H+. Far from the hole the state reduces to the Minkowski vacuum state.

|0K> = Hartle-Hawking-Israel vacuum. Modes are positive frequency wrt the Killing vector of Kruskal time. They are regular on H+. Far from the hole the state reduces to a thermal bath.

"As always in quantum theory, additional physical criteria are necessary to decide which quantum state corresponds to the physical situation of interest. If the universe contains an eternal black hole, only observation can reveal what quantum state is actually realized. However |0K> clearly reproduces the features outside the hole that would be present if a black hole that was formed from collapsing body were subsequently confined in a box and allowed to come into thermal equilibrium."

"A third vacuum, |0U>, the Unruh vacuum, yields a time-asymmetric thermal flux from the hole rather than a thermal bath."